LMFDB - Embedded newform 2151.4.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.323742 q^{2} -7.89519 q^{4} -13.7872 q^{5} +10.6908 q^{7} -5.14594 q^{8} +O(q^{10})$	\(q+0.323742 q^{2} -7.89519 q^{4} -13.7872 q^{5} +10.6908 q^{7} -5.14594 q^{8} -4.46348 q^{10} -6.55234 q^{11} -49.3322 q^{13} +3.46106 q^{14} +61.4956 q^{16} -52.0567 q^{17} +87.4046 q^{19} +108.852 q^{20} -2.12127 q^{22} +101.071 q^{23} +65.0858 q^{25} -15.9709 q^{26} -84.4059 q^{28} -290.605 q^{29} +112.054 q^{31} +61.0762 q^{32} -16.8529 q^{34} -147.396 q^{35} -88.0908 q^{37} +28.2965 q^{38} +70.9479 q^{40} +332.547 q^{41} +376.858 q^{43} +51.7319 q^{44} +32.7208 q^{46} +358.596 q^{47} -228.707 q^{49} +21.0710 q^{50} +389.487 q^{52} -249.863 q^{53} +90.3381 q^{55} -55.0142 q^{56} -94.0810 q^{58} +750.805 q^{59} +798.395 q^{61} +36.2767 q^{62} -472.192 q^{64} +680.151 q^{65} -246.448 q^{67} +410.998 q^{68} -47.7182 q^{70} -943.317 q^{71} +566.499 q^{73} -28.5187 q^{74} -690.076 q^{76} -70.0497 q^{77} +1184.65 q^{79} -847.849 q^{80} +107.659 q^{82} -796.105 q^{83} +717.714 q^{85} +122.005 q^{86} +33.7179 q^{88} -458.059 q^{89} -527.401 q^{91} -797.972 q^{92} +116.093 q^{94} -1205.06 q^{95} +378.798 q^{97} -74.0420 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.323742	0.114460	0.0572300	−	0.998361i	$-0.481773\pi$
$2$	0.323742	0.114460	0.0572300	+	0.998361i	$0.481773\pi$
$3$	0	0
$3$	0	0
$4$	−7.89519	−0.986899
$5$	−13.7872	−1.23316	−0.616581	−	0.787292i	$-0.711483\pi$
$5$	−13.7872	−1.23316	−0.616581	+	0.787292i	$0.711483\pi$
$6$	0	0
$7$	10.6908	0.577249	0.288624	−	0.957442i	$-0.406802\pi$
$7$	10.6908	0.577249	0.288624	+	0.957442i	$0.406802\pi$
$8$	−5.14594	−0.227420
$9$	0	0
$10$	−4.46348	−0.141148
$11$	−6.55234	−0.179600	−0.0898002	−	0.995960i	$-0.528623\pi$
$11$	−6.55234	−0.179600	−0.0898002	+	0.995960i	$0.528623\pi$
$12$	0	0
$13$	−49.3322	−1.05248	−0.526242	−	0.850335i	$-0.676399\pi$
$13$	−49.3322	−1.05248	−0.526242	+	0.850335i	$0.676399\pi$
$14$	3.46106	0.0660719
$15$	0	0
$16$	61.4956	0.960868
$17$	−52.0567	−0.742683	−0.371341	−	0.928496i	$-0.621102\pi$
$17$	−52.0567	−0.742683	−0.371341	+	0.928496i	$0.621102\pi$
$18$	0	0
$19$	87.4046	1.05537	0.527684	−	0.849441i	$-0.323061\pi$
$19$	87.4046	1.05537	0.527684	+	0.849441i	$0.323061\pi$
$20$	108.852	1.21701
$21$	0	0
$22$	−2.12127	−0.0205571
$23$	101.071	0.916291	0.458146	−	0.888877i	$-0.348514\pi$
$23$	101.071	0.916291	0.458146	+	0.888877i	$0.348514\pi$
$24$	0	0
$25$	65.0858	0.520686
$26$	−15.9709	−0.120467
$27$	0	0
$28$	−84.4059	−0.569686
$29$	−290.605	−1.86083	−0.930414	−	0.366510i	$-0.880553\pi$
$29$	−290.605	−1.86083	−0.930414	+	0.366510i	$0.880553\pi$
$30$	0	0
$31$	112.054	0.649212	0.324606	−	0.945849i	$-0.394768\pi$
$31$	112.054	0.649212	0.324606	+	0.945849i	$0.394768\pi$
$32$	61.0762	0.337401
$33$	0	0
$34$	−16.8529	−0.0850075
$35$	−147.396	−0.711841
$36$	0	0
$37$	−88.0908	−0.391406	−0.195703	−	0.980663i	$-0.562699\pi$
$37$	−88.0908	−0.391406	−0.195703	+	0.980663i	$0.562699\pi$
$38$	28.2965	0.120797
$39$	0	0
$40$	70.9479	0.280446
$41$	332.547	1.26671	0.633356	−	0.773861i	$-0.281677\pi$
$41$	332.547	1.26671	0.633356	+	0.773861i	$0.281677\pi$
$42$	0	0
$43$	376.858	1.33652	0.668259	−	0.743929i	$-0.267040\pi$
$43$	376.858	1.33652	0.668259	+	0.743929i	$0.267040\pi$
$44$	51.7319	0.177247
$45$	0	0
$46$	32.7208	0.104879
$47$	358.596	1.11291	0.556453	−	0.830879i	$-0.312162\pi$
$47$	358.596	1.11291	0.556453	+	0.830879i	$0.312162\pi$
$48$	0	0
$49$	−228.707	−0.666784
$50$	21.0710	0.0595978
$51$	0	0
$52$	389.487	1.03870
$53$	−249.863	−0.647573	−0.323787	−	0.946130i	$-0.604956\pi$
$53$	−249.863	−0.647573	−0.323787	+	0.946130i	$0.604956\pi$
$54$	0	0
$55$	90.3381	0.221476
$56$	−55.0142	−0.131278
$57$	0	0
$58$	−94.0810	−0.212990
$59$	750.805	1.65672	0.828360	−	0.560196i	$-0.189274\pi$
$59$	750.805	1.65672	0.828360	+	0.560196i	$0.189274\pi$
$60$	0	0
$61$	798.395	1.67580	0.837902	−	0.545821i	$-0.183782\pi$
$61$	798.395	1.67580	0.837902	+	0.545821i	$0.183782\pi$
$62$	36.2767	0.0743088
$63$	0	0
$64$	−472.192	−0.922249
$65$	680.151	1.29788
$66$	0	0
$67$	−246.448	−0.449380	−0.224690	−	0.974430i	$-0.572137\pi$
$67$	−246.448	−0.449380	−0.224690	+	0.974430i	$0.572137\pi$
$68$	410.998	0.732953
$69$	0	0
$70$	−47.7182	−0.0814773
$71$	−943.317	−1.57678	−0.788388	−	0.615179i	$-0.789084\pi$
$71$	−943.317	−1.57678	−0.788388	+	0.615179i	$0.789084\pi$
$72$	0	0
$73$	566.499	0.908269	0.454135	−	0.890933i	$-0.349949\pi$
$73$	566.499	0.908269	0.454135	+	0.890933i	$0.349949\pi$
$74$	−28.5187	−0.0448004
$75$	0	0
$76$	−690.076	−1.04154
$77$	−70.0497	−0.103674
$78$	0	0
$79$	1184.65	1.68713	0.843565	−	0.537026i	$-0.180452\pi$
$79$	1184.65	1.68713	0.843565	+	0.537026i	$0.180452\pi$
$80$	−847.849	−1.18491
$81$	0	0
$82$	107.659	0.144988
$83$	−796.105	−1.05282	−0.526409	−	0.850231i	$-0.676462\pi$
$83$	−796.105	−1.05282	−0.526409	+	0.850231i	$0.676462\pi$
$84$	0	0
$85$	717.714	0.915847
$86$	122.005	0.152978
$87$	0	0
$88$	33.7179	0.0408448
$89$	−458.059	−0.545552	−0.272776	−	0.962078i	$-0.587942\pi$
$89$	−458.059	−0.545552	−0.272776	+	0.962078i	$0.587942\pi$
$90$	0	0
$91$	−527.401	−0.607545
$92$	−797.972	−0.904287
$93$	0	0
$94$	116.093	0.127383
$95$	−1205.06	−1.30144
$96$	0	0
$97$	378.798	0.396506	0.198253	−	0.980151i	$-0.436473\pi$
$97$	378.798	0.396506	0.198253	+	0.980151i	$0.436473\pi$
$98$	−74.0420	−0.0763201
$99$	0	0
$100$	−513.865	−0.513865
$101$	418.086	0.411892	0.205946	−	0.978563i	$-0.433973\pi$
$101$	418.086	0.411892	0.205946	+	0.978563i	$0.433973\pi$
$102$	0	0
$103$	−44.1303	−0.0422164	−0.0211082	−	0.999777i	$-0.506719\pi$
$103$	−44.1303	−0.0422164	−0.0211082	+	0.999777i	$0.506719\pi$
$104$	253.860	0.239356
$105$	0	0
$106$	−80.8912	−0.0741212
$107$	−724.473	−0.654556	−0.327278	−	0.944928i	$-0.606131\pi$
$107$	−724.473	−0.654556	−0.327278	+	0.944928i	$0.606131\pi$
$108$	0	0
$109$	1134.44	0.996875	0.498438	−	0.866926i	$-0.333907\pi$
$109$	1134.44	0.996875	0.498438	+	0.866926i	$0.333907\pi$
$110$	29.2462	0.0253502
$111$	0	0
$112$	657.437	0.554660
$113$	−1795.45	−1.49470	−0.747352	−	0.664429i	$-0.768675\pi$
$113$	−1795.45	−1.49470	−0.747352	+	0.664429i	$0.768675\pi$
$114$	0	0
$115$	−1393.48	−1.12993
$116$	2294.38	1.83645
$117$	0	0
$118$	243.067	0.189628
$119$	−556.528	−0.428713
$120$	0	0
$121$	−1288.07	−0.967744
$122$	258.474	0.191813
$123$	0	0
$124$	−884.691	−0.640706
$125$	826.047	0.591071
$126$	0	0
$127$	−125.858	−0.0879380	−0.0439690	−	0.999033i	$-0.514000\pi$
$127$	−125.858	−0.0879380	−0.0439690	+	0.999033i	$0.514000\pi$
$128$	−641.478	−0.442962
$129$	0	0
$130$	220.193	0.148556
$131$	209.463	0.139701	0.0698506	−	0.997557i	$-0.477748\pi$
$131$	209.463	0.139701	0.0698506	+	0.997557i	$0.477748\pi$
$132$	0	0
$133$	934.425	0.609210
$134$	−79.7856	−0.0514360
$135$	0	0
$136$	267.881	0.168901
$137$	45.2777	0.0282360	0.0141180	−	0.999900i	$-0.495506\pi$
$137$	45.2777	0.0282360	0.0141180	+	0.999900i	$0.495506\pi$
$138$	0	0
$139$	−1086.95	−0.663263	−0.331632	−	0.943409i	$-0.607599\pi$
$139$	−1086.95	−0.663263	−0.331632	+	0.943409i	$0.607599\pi$
$140$	1163.72	0.702515
$141$	0	0
$142$	−305.391	−0.180478
$143$	323.241	0.189026
$144$	0	0
$145$	4006.62	2.29470
$146$	183.399	0.103960
$147$	0	0
$148$	695.493	0.386278
$149$	389.067	0.213917	0.106958	−	0.994263i	$-0.465889\pi$
$149$	389.067	0.213917	0.106958	+	0.994263i	$0.465889\pi$
$150$	0	0
$151$	−964.818	−0.519972	−0.259986	−	0.965612i	$-0.583718\pi$
$151$	−964.818	−0.519972	−0.259986	+	0.965612i	$0.583718\pi$
$152$	−449.779	−0.240012
$153$	0	0
$154$	−22.6780	−0.0118665
$155$	−1544.91	−0.800583
$156$	0	0
$157$	3085.57	1.56850	0.784252	−	0.620442i	$-0.213047\pi$
$157$	3085.57	1.56850	0.784252	+	0.620442i	$0.213047\pi$
$158$	383.520	0.193109
$159$	0	0
$160$	−842.067	−0.416070
$161$	1080.53	0.528928
$162$	0	0
$163$	1285.68	0.617804	0.308902	−	0.951094i	$-0.400039\pi$
$163$	1285.68	0.617804	0.308902	+	0.951094i	$0.400039\pi$
$164$	−2625.52	−1.25012
$165$	0	0
$166$	−257.733	−0.120506
$167$	−502.689	−0.232929	−0.116465	−	0.993195i	$-0.537156\pi$
$167$	−502.689	−0.232929	−0.116465	+	0.993195i	$0.537156\pi$
$168$	0	0
$169$	236.666	0.107722
$170$	232.354	0.104828
$171$	0	0
$172$	−2975.36	−1.31901
$173$	713.506	0.313565	0.156783	−	0.987633i	$-0.449888\pi$
$173$	713.506	0.313565	0.156783	+	0.987633i	$0.449888\pi$
$174$	0	0
$175$	695.819	0.300566
$176$	−402.940	−0.172572
$177$	0	0
$178$	−148.293	−0.0624439
$179$	−1405.99	−0.587088	−0.293544	−	0.955946i	$-0.594835\pi$
$179$	−1405.99	−0.587088	−0.293544	+	0.955946i	$0.594835\pi$
$180$	0	0
$181$	−4096.73	−1.68236	−0.841182	−	0.540752i	$-0.818140\pi$
$181$	−4096.73	−1.68236	−0.841182	+	0.540752i	$0.818140\pi$
$182$	−170.742	−0.0695396
$183$	0	0
$184$	−520.103	−0.208383
$185$	1214.52	0.482667
$186$	0	0
$187$	341.093	0.133386
$188$	−2831.18	−1.09833
$189$	0	0
$190$	−390.129	−0.148963
$191$	−2683.27	−1.01652	−0.508258	−	0.861205i	$-0.669710\pi$
$191$	−2683.27	−1.01652	−0.508258	+	0.861205i	$0.669710\pi$
$192$	0	0
$193$	1372.03	0.511714	0.255857	−	0.966715i	$-0.417642\pi$
$193$	1372.03	0.511714	0.255857	+	0.966715i	$0.417642\pi$
$194$	122.633	0.0453841
$195$	0	0
$196$	1805.68	0.658048
$197$	−2148.28	−0.776947	−0.388474	−	0.921460i	$-0.626998\pi$
$197$	−2148.28	−0.776947	−0.388474	+	0.921460i	$0.626998\pi$
$198$	0	0
$199$	−3998.60	−1.42439	−0.712194	−	0.701982i	$-0.752299\pi$
$199$	−3998.60	−1.42439	−0.712194	+	0.701982i	$0.752299\pi$
$200$	−334.927	−0.118415
$201$	0	0
$202$	135.352	0.0471452
$203$	−3106.80	−1.07416
$204$	0	0
$205$	−4584.88	−1.56206
$206$	−14.2868	−0.00483209
$207$	0	0
$208$	−3033.71	−1.01130
$209$	−572.704	−0.189544
$210$	0	0
$211$	−3128.52	−1.02074	−0.510369	−	0.859955i	$-0.670491\pi$
$211$	−3128.52	−1.02074	−0.510369	+	0.859955i	$0.670491\pi$
$212$	1972.72	0.639089
$213$	0	0
$214$	−234.542	−0.0749205
$215$	−5195.80	−1.64814
$216$	0	0
$217$	1197.95	0.374757
$218$	367.265	0.114102
$219$	0	0
$220$	−713.237	−0.218575
$221$	2568.07	0.781662
$222$	0	0
$223$	−5203.01	−1.56242	−0.781209	−	0.624270i	$-0.785397\pi$
$223$	−5203.01	−1.56242	−0.781209	+	0.624270i	$0.785397\pi$
$224$	652.953	0.194765
$225$	0	0
$226$	−581.261	−0.171084
$227$	2651.35	0.775225	0.387613	−	0.921822i	$-0.373300\pi$
$227$	2651.35	0.775225	0.387613	+	0.921822i	$0.373300\pi$
$228$	0	0
$229$	−4677.06	−1.34964	−0.674822	−	0.737980i	$-0.735780\pi$
$229$	−4677.06	−1.34964	−0.674822	+	0.737980i	$0.735780\pi$
$230$	−451.127	−0.129332
$231$	0	0
$232$	1495.44	0.423190
$233$	−6155.59	−1.73076	−0.865378	−	0.501120i	$-0.832922\pi$
$233$	−6155.59	−1.73076	−0.865378	+	0.501120i	$0.832922\pi$
$234$	0	0
$235$	−4944.02	−1.37239
$236$	−5927.75	−1.63501
$237$	0	0
$238$	−180.171	−0.0490705
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	5789.31	1.54740	0.773698	−	0.633555i	$-0.218405\pi$
$241$	5789.31	1.54740	0.773698	+	0.633555i	$0.218405\pi$
$242$	−417.001	−0.110768
$243$	0	0
$244$	−6303.48	−1.65385
$245$	3153.22	0.822252
$246$	0	0
$247$	−4311.86	−1.11076
$248$	−576.625	−0.147644
$249$	0	0
$250$	267.426	0.0676540
$251$	5788.45	1.45563	0.727816	−	0.685772i	$-0.240535\pi$
$251$	5788.45	1.45563	0.727816	+	0.685772i	$0.240535\pi$
$252$	0	0
$253$	−662.249	−0.164566
$254$	−40.7456	−0.0100654
$255$	0	0
$256$	3569.86	0.871548
$257$	3821.16	0.927460	0.463730	−	0.885977i	$-0.346511\pi$
$257$	3821.16	0.927460	0.463730	+	0.885977i	$0.346511\pi$
$258$	0	0
$259$	−941.760	−0.225939
$260$	−5369.92	−1.28088
$261$	0	0
$262$	67.8119	0.0159902
$263$	−6665.94	−1.56289	−0.781444	−	0.623976i	$-0.785516\pi$
$263$	−6665.94	−1.56289	−0.781444	+	0.623976i	$0.785516\pi$
$264$	0	0
$265$	3444.91	0.798562
$266$	302.512	0.0697302
$267$	0	0
$268$	1945.76	0.443493
$269$	6719.19	1.52296	0.761481	−	0.648188i	$-0.224473\pi$
$269$	6719.19	1.52296	0.761481	+	0.648188i	$0.224473\pi$
$270$	0	0
$271$	4476.85	1.00350	0.501752	−	0.865012i	$-0.332689\pi$
$271$	4476.85	1.00350	0.501752	+	0.865012i	$0.332689\pi$
$272$	−3201.26	−0.713620
$273$	0	0
$274$	14.6583	0.00323189
$275$	−426.464	−0.0935155
$276$	0	0
$277$	−6667.59	−1.44627	−0.723135	−	0.690707i	$-0.757299\pi$
$277$	−6667.59	−1.44627	−0.723135	+	0.690707i	$0.757299\pi$
$278$	−351.890	−0.0759171
$279$	0	0
$280$	758.489	0.161887
$281$	−8183.71	−1.73736	−0.868682	−	0.495370i	$-0.835032\pi$
$281$	−8183.71	−1.73736	−0.868682	+	0.495370i	$0.835032\pi$
$282$	0	0
$283$	−2828.09	−0.594037	−0.297019	−	0.954872i	$-0.595992\pi$
$283$	−2828.09	−0.594037	−0.297019	+	0.954872i	$0.595992\pi$
$284$	7447.66	1.55612
$285$	0	0
$286$	104.647	0.0216360
$287$	3555.20	0.731208
$288$	0	0
$289$	−2203.10	−0.448422
$290$	1297.11	0.262652
$291$	0	0
$292$	−4472.61	−0.896370
$293$	6430.97	1.28226	0.641129	−	0.767434i	$-0.278467\pi$
$293$	6430.97	1.28226	0.641129	+	0.767434i	$0.278467\pi$
$294$	0	0
$295$	−10351.5	−2.04300
$296$	453.310	0.0890138
$297$	0	0
$298$	125.957	0.0244849
$299$	−4986.04	−0.964382
$300$	0	0
$301$	4028.91	0.771503
$302$	−312.352	−0.0595160
$303$	0	0
$304$	5375.00	1.01407
$305$	−11007.6	−2.06654
$306$	0	0
$307$	228.417	0.0424640	0.0212320	−	0.999775i	$-0.493241\pi$
$307$	228.417	0.0424640	0.0212320	+	0.999775i	$0.493241\pi$
$308$	553.056	0.102316
$309$	0	0
$310$	−500.153	−0.0916347
$311$	1404.38	0.256062	0.128031	−	0.991770i	$-0.459134\pi$
$311$	1404.38	0.256062	0.128031	+	0.991770i	$0.459134\pi$
$312$	0	0
$313$	−10465.2	−1.88986	−0.944931	−	0.327269i	$-0.893872\pi$
$313$	−10465.2	−1.88986	−0.944931	+	0.327269i	$0.893872\pi$
$314$	998.928	0.179531
$315$	0	0
$316$	−9353.02	−1.66503
$317$	2882.53	0.510723	0.255362	−	0.966846i	$-0.417805\pi$
$317$	2882.53	0.510723	0.255362	+	0.966846i	$0.417805\pi$
$318$	0	0
$319$	1904.14	0.334205
$320$	6510.18	1.13728
$321$	0	0
$322$	349.811	0.0605411
$323$	−4549.99	−0.783803
$324$	0	0
$325$	−3210.83	−0.548014
$326$	416.227	0.0707138
$327$	0	0
$328$	−1711.27	−0.288076
$329$	3833.68	0.642424
$330$	0	0
$331$	7513.09	1.24760	0.623802	−	0.781583i	$-0.285587\pi$
$331$	7513.09	1.24760	0.623802	+	0.781583i	$0.285587\pi$
$332$	6285.40	1.03902
$333$	0	0
$334$	−162.741	−0.0266611
$335$	3397.82	0.554158
$336$	0	0
$337$	950.978	0.153718	0.0768592	−	0.997042i	$-0.475511\pi$
$337$	950.978	0.153718	0.0768592	+	0.997042i	$0.475511\pi$
$338$	76.6186	0.0123299
$339$	0	0
$340$	−5666.49	−0.903849
$341$	−734.218	−0.116599
$342$	0	0
$343$	−6112.00	−0.962149
$344$	−1939.29	−0.303951
$345$	0	0
$346$	230.992	0.0358907
$347$	4191.68	0.648476	0.324238	−	0.945976i	$-0.394892\pi$
$347$	4191.68	0.648476	0.324238	+	0.945976i	$0.394892\pi$
$348$	0	0
$349$	−7801.82	−1.19662	−0.598312	−	0.801263i	$-0.704162\pi$
$349$	−7801.82	−1.19662	−0.598312	+	0.801263i	$0.704162\pi$
$350$	225.266	0.0344027
$351$	0	0
$352$	−400.192	−0.0605974
$353$	9743.49	1.46910	0.734552	−	0.678553i	$-0.237392\pi$
$353$	9743.49	1.46910	0.734552	+	0.678553i	$0.237392\pi$
$354$	0	0
$355$	13005.7	1.94442
$356$	3616.46	0.538405
$357$	0	0
$358$	−455.178	−0.0671981
$359$	−2092.85	−0.307677	−0.153839	−	0.988096i	$-0.549164\pi$
$359$	−2092.85	−0.307677	−0.153839	+	0.988096i	$0.549164\pi$
$360$	0	0
$361$	780.561	0.113801
$362$	−1326.28	−0.192563
$363$	0	0
$364$	4163.93	0.599586
$365$	−7810.41	−1.12004
$366$	0	0
$367$	−1089.05	−0.154898	−0.0774492	−	0.996996i	$-0.524678\pi$
$367$	−1089.05	−0.154898	−0.0774492	+	0.996996i	$0.524678\pi$
$368$	6215.40	0.880435
$369$	0	0
$370$	393.191	0.0552461
$371$	−2671.24	−0.373811
$372$	0	0
$373$	2411.61	0.334768	0.167384	−	0.985892i	$-0.446468\pi$
$373$	2411.61	0.334768	0.167384	+	0.985892i	$0.446468\pi$
$374$	110.426	0.0152674
$375$	0	0
$376$	−1845.31	−0.253098
$377$	14336.2	1.95849
$378$	0	0
$379$	−6170.09	−0.836243	−0.418121	−	0.908391i	$-0.637311\pi$
$379$	−6170.09	−0.836243	−0.418121	+	0.908391i	$0.637311\pi$
$380$	9514.19	1.28439
$381$	0	0
$382$	−868.686	−0.116350
$383$	7587.95	1.01234	0.506170	−	0.862434i	$-0.331061\pi$
$383$	7587.95	1.01234	0.506170	+	0.862434i	$0.331061\pi$
$384$	0	0
$385$	965.786	0.127847
$386$	444.183	0.0585707
$387$	0	0
$388$	−2990.68	−0.391311
$389$	−518.713	−0.0676087	−0.0338044	−	0.999428i	$-0.510762\pi$
$389$	−518.713	−0.0676087	−0.0338044	+	0.999428i	$0.510762\pi$
$390$	0	0
$391$	−5261.41	−0.680513
$392$	1176.91	0.151640
$393$	0	0
$394$	−695.488	−0.0889294
$395$	−16332.9	−2.08050
$396$	0	0
$397$	8742.75	1.10525	0.552627	−	0.833429i	$-0.313625\pi$
$397$	8742.75	1.10525	0.552627	+	0.833429i	$0.313625\pi$
$398$	−1294.51	−0.163036
$399$	0	0
$400$	4002.49	0.500311
$401$	−3525.27	−0.439012	−0.219506	−	0.975611i	$-0.570445\pi$
$401$	−3525.27	−0.439012	−0.219506	+	0.975611i	$0.570445\pi$
$402$	0	0
$403$	−5527.89	−0.683285
$404$	−3300.87	−0.406496
$405$	0	0
$406$	−1005.80	−0.122948
$407$	577.200	0.0702967
$408$	0	0
$409$	−13231.4	−1.59964	−0.799820	−	0.600240i	$-0.795072\pi$
$409$	−13231.4	−1.59964	−0.799820	+	0.600240i	$0.795072\pi$
$410$	−1484.32	−0.178793
$411$	0	0
$412$	348.417	0.0416633
$413$	8026.70	0.956339
$414$	0	0
$415$	10976.0	1.29829
$416$	−3013.02	−0.355110
$417$	0	0
$418$	−185.408	−0.0216953
$419$	−11151.8	−1.30025	−0.650123	−	0.759829i	$-0.725282\pi$
$419$	−11151.8	−1.30025	−0.650123	+	0.759829i	$0.725282\pi$
$420$	0	0
$421$	−5786.20	−0.669838	−0.334919	−	0.942247i	$-0.608709\pi$
$421$	−5786.20	−0.669838	−0.334919	+	0.942247i	$0.608709\pi$
$422$	−1012.83	−0.116834
$423$	0	0
$424$	1285.78	0.147271
$425$	−3388.15	−0.386705
$426$	0	0
$427$	8535.48	0.967356
$428$	5719.85	0.645980
$429$	0	0
$430$	−1682.10	−0.188646
$431$	−8584.90	−0.959443	−0.479722	−	0.877421i	$-0.659262\pi$
$431$	−8584.90	−0.959443	−0.479722	+	0.877421i	$0.659262\pi$
$432$	0	0
$433$	−1059.97	−0.117642	−0.0588208	−	0.998269i	$-0.518734\pi$
$433$	−1059.97	−0.117642	−0.0588208	+	0.998269i	$0.518734\pi$
$434$	387.827	0.0428947
$435$	0	0
$436$	−8956.60	−0.983815
$437$	8834.04	0.967024
$438$	0	0
$439$	−6027.23	−0.655271	−0.327635	−	0.944804i	$-0.606252\pi$
$439$	−6027.23	−0.655271	−0.327635	+	0.944804i	$0.606252\pi$
$440$	−464.874	−0.0503682
$441$	0	0
$442$	831.392	0.0894690
$443$	10419.0	1.11743	0.558715	−	0.829360i	$-0.311295\pi$
$443$	10419.0	1.11743	0.558715	+	0.829360i	$0.311295\pi$
$444$	0	0
$445$	6315.33	0.672754
$446$	−1684.43	−0.178834
$447$	0	0
$448$	−5048.11	−0.532367
$449$	18085.1	1.90086	0.950432	−	0.310933i	$-0.100642\pi$
$449$	18085.1	1.90086	0.950432	+	0.310933i	$0.100642\pi$
$450$	0	0
$451$	−2178.96	−0.227502
$452$	14175.4	1.47512
$453$	0	0
$454$	858.352	0.0887323
$455$	7271.36	0.749201
$456$	0	0
$457$	−18232.5	−1.86626	−0.933130	−	0.359539i	$-0.882934\pi$
$457$	−18232.5	−1.86626	−0.933130	+	0.359539i	$0.882934\pi$
$458$	−1514.16	−0.154480
$459$	0	0
$460$	11001.8	1.11513
$461$	11754.0	1.18750	0.593751	−	0.804649i	$-0.297646\pi$
$461$	11754.0	1.18750	0.593751	+	0.804649i	$0.297646\pi$
$462$	0	0
$463$	6161.15	0.618430	0.309215	−	0.950992i	$-0.399934\pi$
$463$	6161.15	0.618430	0.309215	+	0.950992i	$0.399934\pi$
$464$	−17870.9	−1.78801
$465$	0	0
$466$	−1992.82	−0.198102
$467$	−16575.8	−1.64247	−0.821237	−	0.570587i	$-0.806716\pi$
$467$	−16575.8	−1.64247	−0.821237	+	0.570587i	$0.806716\pi$
$468$	0	0
$469$	−2634.73	−0.259404
$470$	−1600.59	−0.157084
$471$	0	0
$472$	−3863.60	−0.376772
$473$	−2469.30	−0.240039
$474$	0	0
$475$	5688.80	0.549516
$476$	4393.89	0.423096
$477$	0	0
$478$	−77.3743	−0.00740380
$479$	−353.652	−0.0337344	−0.0168672	−	0.999858i	$-0.505369\pi$
$479$	−353.652	−0.0337344	−0.0168672	+	0.999858i	$0.505369\pi$
$480$	0	0
$481$	4345.71	0.411949
$482$	1874.24	0.177115
$483$	0	0
$484$	10169.5	0.955065
$485$	−5222.54	−0.488956
$486$	0	0
$487$	13574.6	1.26309	0.631545	−	0.775339i	$-0.282421\pi$
$487$	13574.6	1.26309	0.631545	+	0.775339i	$0.282421\pi$
$488$	−4108.49	−0.381112
$489$	0	0
$490$	1020.83	0.0941150
$491$	9683.26	0.890019	0.445010	−	0.895526i	$-0.353200\pi$
$491$	9683.26	0.890019	0.445010	+	0.895526i	$0.353200\pi$
$492$	0	0
$493$	15127.9	1.38200
$494$	−1395.93	−0.127137
$495$	0	0
$496$	6890.85	0.623807
$497$	−10084.8	−0.910192
$498$	0	0
$499$	812.818	0.0729193	0.0364597	−	0.999335i	$-0.488392\pi$
$499$	812.818	0.0729193	0.0364597	+	0.999335i	$0.488392\pi$
$500$	−6521.80	−0.583327
$501$	0	0
$502$	1873.96	0.166612
$503$	10061.0	0.891844	0.445922	−	0.895072i	$-0.352876\pi$
$503$	10061.0	0.891844	0.445922	+	0.895072i	$0.352876\pi$
$504$	0	0
$505$	−5764.22	−0.507930
$506$	−214.398	−0.0188362
$507$	0	0
$508$	993.677	0.0867860
$509$	5485.16	0.477653	0.238826	−	0.971062i	$-0.423237\pi$
$509$	5485.16	0.477653	0.238826	+	0.971062i	$0.423237\pi$
$510$	0	0
$511$	6056.32	0.524297
$512$	6287.53	0.542720
$513$	0	0
$514$	1237.07	0.106157
$515$	608.432	0.0520596
$516$	0	0
$517$	−2349.64	−0.199878
$518$	−304.887	−0.0258610
$519$	0	0
$520$	−3500.01	−0.295165
$521$	−9628.30	−0.809642	−0.404821	−	0.914396i	$-0.632666\pi$
$521$	−9628.30	−0.809642	−0.404821	+	0.914396i	$0.632666\pi$
$522$	0	0
$523$	−2377.93	−0.198814	−0.0994071	−	0.995047i	$-0.531695\pi$
$523$	−2377.93	−0.198814	−0.0994071	+	0.995047i	$0.531695\pi$
$524$	−1653.75	−0.137871
$525$	0	0
$526$	−2158.04	−0.178888
$527$	−5833.18	−0.482158
$528$	0	0
$529$	−1951.72	−0.160411
$530$	1115.26	0.0914034
$531$	0	0
$532$	−7377.46	−0.601228
$533$	−16405.3	−1.33319
$534$	0	0
$535$	9988.43	0.807173
$536$	1268.21	0.102198
$537$	0	0
$538$	2175.28	0.174318
$539$	1498.56	0.119755
$540$	0	0
$541$	15856.4	1.26011	0.630055	−	0.776550i	$-0.283032\pi$
$541$	15856.4	1.26011	0.630055	+	0.776550i	$0.283032\pi$
$542$	1449.34	0.114861
$543$	0	0
$544$	−3179.43	−0.250582
$545$	−15640.7	−1.22931
$546$	0	0
$547$	−2603.62	−0.203515	−0.101758	−	0.994809i	$-0.532447\pi$
$547$	−2603.62	−0.203515	−0.101758	+	0.994809i	$0.532447\pi$
$548$	−357.476	−0.0278661
$549$	0	0
$550$	−138.064	−0.0107038
$551$	−25400.2	−1.96386
$552$	0	0
$553$	12664.8	0.973894
$554$	−2158.58	−0.165540
$555$	0	0
$556$	8581.65	0.654574
$557$	−18339.7	−1.39511	−0.697555	−	0.716531i	$-0.745729\pi$
$557$	−18339.7	−1.39511	−0.697555	+	0.716531i	$0.745729\pi$
$558$	0	0
$559$	−18591.2	−1.40666
$560$	−9064.19	−0.683985
$561$	0	0
$562$	−2649.41	−0.198859
$563$	−574.638	−0.0430162	−0.0215081	−	0.999769i	$-0.506847\pi$
$563$	−574.638	−0.0430162	−0.0215081	+	0.999769i	$0.506847\pi$
$564$	0	0
$565$	24754.1	1.84321
$566$	−915.571	−0.0679935
$567$	0	0
$568$	4854.25	0.358591
$569$	7487.91	0.551686	0.275843	−	0.961203i	$-0.411043\pi$
$569$	7487.91	0.551686	0.275843	+	0.961203i	$0.411043\pi$
$570$	0	0
$571$	11961.4	0.876650	0.438325	−	0.898817i	$-0.355572\pi$
$571$	11961.4	0.876650	0.438325	+	0.898817i	$0.355572\pi$
$572$	−2552.05	−0.186550
$573$	0	0
$574$	1150.97	0.0836940
$575$	6578.27	0.477100
$576$	0	0
$577$	12429.5	0.896786	0.448393	−	0.893837i	$-0.351997\pi$
$577$	12429.5	0.896786	0.448393	+	0.893837i	$0.351997\pi$
$578$	−713.235	−0.0513264
$579$	0	0
$580$	−31633.0	−2.26464
$581$	−8511.00	−0.607738
$582$	0	0
$583$	1637.19	0.116304
$584$	−2915.17	−0.206559
$585$	0	0
$586$	2081.97	0.146767
$587$	−22408.3	−1.57562	−0.787812	−	0.615916i	$-0.788786\pi$
$587$	−22408.3	−1.57562	−0.787812	+	0.615916i	$0.788786\pi$
$588$	0	0
$589$	9794.07	0.685157
$590$	−3351.20	−0.233842
$591$	0	0
$592$	−5417.19	−0.376090
$593$	−13338.1	−0.923662	−0.461831	−	0.886968i	$-0.652807\pi$
$593$	−13338.1	−0.923662	−0.461831	+	0.886968i	$0.652807\pi$
$594$	0	0
$595$	7672.94	0.528672
$596$	−3071.76	−0.211114
$597$	0	0
$598$	−1614.19	−0.110383
$599$	−4590.11	−0.313100	−0.156550	−	0.987670i	$-0.550037\pi$
$599$	−4590.11	−0.313100	−0.156550	+	0.987670i	$0.550037\pi$
$600$	0	0
$601$	−9058.35	−0.614805	−0.307402	−	0.951580i	$-0.599460\pi$
$601$	−9058.35	−0.614805	−0.307402	+	0.951580i	$0.599460\pi$
$602$	1304.33	0.0883063
$603$	0	0
$604$	7617.42	0.513160
$605$	17758.8	1.19338
$606$	0	0
$607$	538.222	0.0359897	0.0179949	−	0.999838i	$-0.494272\pi$
$607$	538.222	0.0359897	0.0179949	+	0.999838i	$0.494272\pi$
$608$	5338.34	0.356083
$609$	0	0
$610$	−3563.62	−0.236536
$611$	−17690.3	−1.17132
$612$	0	0
$613$	25656.1	1.69044	0.845220	−	0.534418i	$-0.179469\pi$
$613$	25656.1	1.69044	0.845220	+	0.534418i	$0.179469\pi$
$614$	73.9482	0.00486043
$615$	0	0
$616$	360.471	0.0235776
$617$	−2149.29	−0.140239	−0.0701194	−	0.997539i	$-0.522338\pi$
$617$	−2149.29	−0.140239	−0.0701194	+	0.997539i	$0.522338\pi$
$618$	0	0
$619$	−26908.6	−1.74725	−0.873624	−	0.486602i	$-0.838236\pi$
$619$	−26908.6	−1.74725	−0.873624	+	0.486602i	$0.838236\pi$
$620$	12197.4	0.790094
$621$	0	0
$622$	454.658	0.0293089
$623$	−4897.01	−0.314919
$624$	0	0
$625$	−19524.6	−1.24957
$626$	−3388.02	−0.216314
$627$	0	0
$628$	−24361.2	−1.54796
$629$	4585.72	0.290691
$630$	0	0
$631$	−15723.0	−0.991953	−0.495977	−	0.868336i	$-0.665190\pi$
$631$	−15723.0	−0.991953	−0.495977	+	0.868336i	$0.665190\pi$
$632$	−6096.13	−0.383688
$633$	0	0
$634$	933.197	0.0584574
$635$	1735.23	0.108442
$636$	0	0
$637$	11282.6	0.701779
$638$	616.451	0.0382531
$639$	0	0
$640$	8844.16	0.546244
$641$	−27327.9	−1.68391	−0.841956	−	0.539546i	$-0.818596\pi$
$641$	−27327.9	−1.68391	−0.841956	+	0.539546i	$0.818596\pi$
$642$	0	0
$643$	30867.5	1.89315	0.946574	−	0.322488i	$-0.104519\pi$
$643$	30867.5	1.89315	0.946574	+	0.322488i	$0.104519\pi$
$644$	−8530.96	−0.521998
$645$	0	0
$646$	−1473.02	−0.0897141
$647$	11652.3	0.708038	0.354019	−	0.935238i	$-0.384815\pi$
$647$	11652.3	0.708038	0.354019	+	0.935238i	$0.384815\pi$
$648$	0	0
$649$	−4919.53	−0.297547
$650$	−1039.48	−0.0627257
$651$	0	0
$652$	−10150.7	−0.609710
$653$	18018.5	1.07981	0.539906	−	0.841725i	$-0.318460\pi$
$653$	18018.5	1.07981	0.539906	+	0.841725i	$0.318460\pi$
$654$	0	0
$655$	−2887.90	−0.172274
$656$	20450.2	1.21714
$657$	0	0
$658$	1241.12	0.0735319
$659$	27157.6	1.60533	0.802663	−	0.596433i	$-0.203416\pi$
$659$	27157.6	1.60533	0.802663	+	0.596433i	$0.203416\pi$
$660$	0	0
$661$	19621.2	1.15458	0.577290	−	0.816539i	$-0.304110\pi$
$661$	19621.2	1.15458	0.577290	+	0.816539i	$0.304110\pi$
$662$	2432.30	0.142801
$663$	0	0
$664$	4096.71	0.239432
$665$	−12883.1	−0.751254
$666$	0	0
$667$	−29371.7	−1.70506
$668$	3968.82	0.229878
$669$	0	0
$670$	1100.02	0.0634289
$671$	−5231.35	−0.300975
$672$	0	0
$673$	1521.83	0.0871652	0.0435826	−	0.999050i	$-0.486123\pi$
$673$	1521.83	0.0871652	0.0435826	+	0.999050i	$0.486123\pi$
$674$	307.871	0.0175946
$675$	0	0
$676$	−1868.52	−0.106311
$677$	27835.7	1.58023	0.790113	−	0.612961i	$-0.210022\pi$
$677$	27835.7	1.58023	0.790113	+	0.612961i	$0.210022\pi$
$678$	0	0
$679$	4049.65	0.228883
$680$	−3693.31	−0.208282
$681$	0	0
$682$	−237.697	−0.0133459
$683$	−11483.3	−0.643334	−0.321667	−	0.946853i	$-0.604243\pi$
$683$	−11483.3	−0.643334	−0.321667	+	0.946853i	$0.604243\pi$
$684$	0	0
$685$	−624.250	−0.0348195
$686$	−1978.71	−0.110128
$687$	0	0
$688$	23175.1	1.28422
$689$	12326.3	0.681560
$690$	0	0
$691$	−34379.2	−1.89268	−0.946342	−	0.323167i	$-0.895253\pi$
$691$	−34379.2	−1.89268	−0.946342	+	0.323167i	$0.895253\pi$
$692$	−5633.26	−0.309457
$693$	0	0
$694$	1357.02	0.0742245
$695$	14985.9	0.817910
$696$	0	0
$697$	−17311.3	−0.940765
$698$	−2525.78	−0.136966
$699$	0	0
$700$	−5493.63	−0.296628
$701$	−3907.45	−0.210531	−0.105265	−	0.994444i	$-0.533569\pi$
$701$	−3907.45	−0.210531	−0.105265	+	0.994444i	$0.533569\pi$
$702$	0	0
$703$	−7699.54	−0.413078
$704$	3093.96	0.165636
$705$	0	0
$706$	3154.37	0.168154
$707$	4469.67	0.237764
$708$	0	0
$709$	19389.8	1.02708	0.513539	−	0.858066i	$-0.328334\pi$
$709$	19389.8	1.02708	0.513539	+	0.858066i	$0.328334\pi$
$710$	4210.47	0.222558
$711$	0	0
$712$	2357.14	0.124070
$713$	11325.4	0.594867
$714$	0	0
$715$	−4456.58	−0.233100
$716$	11100.6	0.579397
$717$	0	0
$718$	−677.542	−0.0352168
$719$	−13085.4	−0.678727	−0.339363	−	0.940655i	$-0.610212\pi$
$719$	−13085.4	−0.678727	−0.339363	+	0.940655i	$0.610212\pi$
$720$	0	0
$721$	−471.788	−0.0243694
$722$	252.700	0.0130257
$723$	0	0
$724$	32344.5	1.66032
$725$	−18914.3	−0.968908
$726$	0	0
$727$	−6981.26	−0.356149	−0.178075	−	0.984017i	$-0.556987\pi$
$727$	−6981.26	−0.356149	−0.178075	+	0.984017i	$0.556987\pi$
$728$	2713.97	0.138168
$729$	0	0
$730$	−2528.56	−0.128200
$731$	−19618.0	−0.992609
$732$	0	0
$733$	−23632.5	−1.19084	−0.595420	−	0.803415i	$-0.703014\pi$
$733$	−23632.5	−1.19084	−0.595420	+	0.803415i	$0.703014\pi$
$734$	−352.570	−0.0177297
$735$	0	0
$736$	6173.01	0.309158
$737$	1614.81	0.0807088
$738$	0	0
$739$	−8326.27	−0.414461	−0.207231	−	0.978292i	$-0.566445\pi$
$739$	−8326.27	−0.414461	−0.207231	+	0.978292i	$0.566445\pi$
$740$	−9588.88	−0.476344
$741$	0	0
$742$	−864.792	−0.0427864
$743$	5098.79	0.251758	0.125879	−	0.992046i	$-0.459825\pi$
$743$	5098.79	0.251758	0.125879	+	0.992046i	$0.459825\pi$
$744$	0	0
$745$	−5364.13	−0.263794
$746$	780.739	0.0383175
$747$	0	0
$748$	−2692.99	−0.131639
$749$	−7745.20	−0.377842
$750$	0	0
$751$	14100.1	0.685111	0.342555	−	0.939498i	$-0.388708\pi$
$751$	14100.1	0.685111	0.342555	+	0.939498i	$0.388708\pi$
$752$	22052.1	1.06936
$753$	0	0
$754$	4641.22	0.224169
$755$	13302.1	0.641209
$756$	0	0
$757$	730.798	0.0350876	0.0175438	−	0.999846i	$-0.494415\pi$
$757$	730.798	0.0350876	0.0175438	+	0.999846i	$0.494415\pi$
$758$	−1997.51	−0.0957164
$759$	0	0
$760$	6201.17	0.295974
$761$	1663.45	0.0792381	0.0396190	−	0.999215i	$-0.487386\pi$
$761$	1663.45	0.0792381	0.0396190	+	0.999215i	$0.487386\pi$
$762$	0	0
$763$	12128.0	0.575445
$764$	21184.9	1.00320
$765$	0	0
$766$	2456.54	0.115872
$767$	−37038.9	−1.74367
$768$	0	0
$769$	−30059.1	−1.40957	−0.704785	−	0.709421i	$-0.748957\pi$
$769$	−30059.1	−1.40957	−0.704785	+	0.709421i	$0.748957\pi$
$770$	312.665	0.0146334
$771$	0	0
$772$	−10832.4	−0.505010
$773$	28145.5	1.30960	0.654802	−	0.755800i	$-0.272752\pi$
$773$	28145.5	1.30960	0.654802	+	0.755800i	$0.272752\pi$
$774$	0	0
$775$	7293.15	0.338036
$776$	−1949.27	−0.0901735
$777$	0	0
$778$	−167.929	−0.00773850
$779$	29066.2	1.33685
$780$	0	0
$781$	6180.93	0.283189
$782$	−1703.34	−0.0778916
$783$	0	0
$784$	−14064.5	−0.640691
$785$	−42541.2	−1.93422
$786$	0	0
$787$	−966.619	−0.0437817	−0.0218909	−	0.999760i	$-0.506969\pi$
$787$	−966.619	−0.0437817	−0.0218909	+	0.999760i	$0.506969\pi$
$788$	16961.1	0.766768
$789$	0	0
$790$	−5287.65	−0.238135
$791$	−19194.8	−0.862816
$792$	0	0
$793$	−39386.6	−1.76376
$794$	2830.39	0.126507
$795$	0	0
$796$	31569.7	1.40573
$797$	−19327.7	−0.859000	−0.429500	−	0.903067i	$-0.641310\pi$
$797$	−19327.7	−0.859000	−0.429500	+	0.903067i	$0.641310\pi$
$798$	0	0
$799$	−18667.3	−0.826537
$800$	3975.19	0.175680
$801$	0	0
$802$	−1141.28	−0.0502493
$803$	−3711.89	−0.163125
$804$	0	0
$805$	−14897.4	−0.652253
$806$	−1789.61	−0.0782088
$807$	0	0
$808$	−2151.45	−0.0936728
$809$	−13191.8	−0.573300	−0.286650	−	0.958035i	$-0.592542\pi$
$809$	−13191.8	−0.573300	−0.286650	+	0.958035i	$0.592542\pi$
$810$	0	0
$811$	12887.0	0.557984	0.278992	−	0.960293i	$-0.410000\pi$
$811$	12887.0	0.557984	0.278992	+	0.960293i	$0.410000\pi$
$812$	24528.8	1.06009
$813$	0	0
$814$	186.864	0.00804616
$815$	−17725.8	−0.761852
$816$	0	0
$817$	32939.1	1.41052
$818$	−4283.57	−0.183095
$819$	0	0
$820$	36198.5	1.54159
$821$	7163.78	0.304528	0.152264	−	0.988340i	$-0.451344\pi$
$821$	7163.78	0.304528	0.152264	+	0.988340i	$0.451344\pi$
$822$	0	0
$823$	−19427.8	−0.822857	−0.411429	−	0.911442i	$-0.634970\pi$
$823$	−19427.8	−0.822857	−0.411429	+	0.911442i	$0.634970\pi$
$824$	227.092	0.00960087
$825$	0	0
$826$	2598.58	0.109463
$827$	31771.9	1.33594	0.667968	−	0.744190i	$-0.267164\pi$
$827$	31771.9	1.33594	0.667968	+	0.744190i	$0.267164\pi$
$828$	0	0
$829$	−23247.3	−0.973960	−0.486980	−	0.873413i	$-0.661902\pi$
$829$	−23247.3	−0.973960	−0.486980	+	0.873413i	$0.661902\pi$
$830$	3553.40	0.148603
$831$	0	0
$832$	23294.3	0.970653
$833$	11905.7	0.495209
$834$	0	0
$835$	6930.65	0.287239
$836$	4521.61	0.187061
$837$	0	0
$838$	−3610.31	−0.148826
$839$	−27865.3	−1.14662	−0.573312	−	0.819337i	$-0.694342\pi$
$839$	−27865.3	−1.14662	−0.573312	+	0.819337i	$0.694342\pi$
$840$	0	0
$841$	60062.3	2.46268
$842$	−1873.23	−0.0766697
$843$	0	0
$844$	24700.2	1.00737
$845$	−3262.95	−0.132839
$846$	0	0
$847$	−13770.5	−0.558629
$848$	−15365.5	−0.622233
$849$	0	0
$850$	−1096.89	−0.0442622
$851$	−8903.39	−0.358642
$852$	0	0
$853$	−38470.7	−1.54421	−0.772106	−	0.635494i	$-0.780797\pi$
$853$	−38470.7	−1.54421	−0.772106	+	0.635494i	$0.780797\pi$
$854$	2763.29	0.110724
$855$	0	0
$856$	3728.09	0.148859
$857$	−32190.2	−1.28308	−0.641538	−	0.767092i	$-0.721703\pi$
$857$	−32190.2	−1.28308	−0.641538	+	0.767092i	$0.721703\pi$
$858$	0	0
$859$	−38.2762	−0.00152034	−0.000760168	−	1.00000i	$-0.500242\pi$
$859$	−38.2762	−0.00152034	−0.000760168		1.00000i	$0.500242\pi$
$860$	41021.8	1.62655
$861$	0	0
$862$	−2779.29	−0.109818
$863$	22784.9	0.898733	0.449367	−	0.893347i	$-0.351650\pi$
$863$	22784.9	0.898733	0.449367	+	0.893347i	$0.351650\pi$
$864$	0	0
$865$	−9837.22	−0.386677
$866$	−343.156	−0.0134653
$867$	0	0
$868$	−9458.05	−0.369847
$869$	−7762.21	−0.303009
$870$	0	0
$871$	12157.8	0.472965
$872$	−5837.74	−0.226710
$873$	0	0
$874$	2859.95	0.110686
$875$	8831.10	0.341195
$876$	0	0
$877$	29423.8	1.13292	0.566460	−	0.824089i	$-0.308313\pi$
$877$	29423.8	1.13292	0.566460	+	0.824089i	$0.308313\pi$
$878$	−1951.27	−0.0750023
$879$	0	0
$880$	5555.39	0.212809
$881$	371.694	0.0142142	0.00710709	−	0.999975i	$-0.497738\pi$
$881$	371.694	0.0142142	0.00710709	+	0.999975i	$0.497738\pi$
$882$	0	0
$883$	10172.2	0.387682	0.193841	−	0.981033i	$-0.437905\pi$
$883$	10172.2	0.387682	0.193841	+	0.981033i	$0.437905\pi$
$884$	−20275.4	−0.771421
$885$	0	0
$886$	3373.07	0.127901
$887$	−38494.1	−1.45716	−0.728582	−	0.684959i	$-0.759820\pi$
$887$	−38494.1	−1.45716	−0.728582	+	0.684959i	$0.759820\pi$
$888$	0	0
$889$	−1345.53	−0.0507621
$890$	2044.54	0.0770034
$891$	0	0
$892$	41078.7	1.54195
$893$	31342.9	1.17453
$894$	0	0
$895$	19384.6	0.723974
$896$	−6857.91	−0.255699
$897$	0	0
$898$	5854.90	0.217573
$899$	−32563.6	−1.20807
$900$	0	0
$901$	13007.1	0.480941
$902$	−705.421	−0.0260399
$903$	0	0
$904$	9239.26	0.339926
$905$	56482.3	2.07463
$906$	0	0
$907$	6939.99	0.254067	0.127033	−	0.991898i	$-0.459454\pi$
$907$	6939.99	0.254067	0.127033	+	0.991898i	$0.459454\pi$
$908$	−20932.9	−0.765069
$909$	0	0
$910$	2354.04	0.0857535
$911$	44169.4	1.60636	0.803181	−	0.595734i	$-0.203139\pi$
$911$	44169.4	1.60636	0.803181	+	0.595734i	$0.203139\pi$
$912$	0	0
$913$	5216.35	0.189086
$914$	−5902.63	−0.213612
$915$	0	0
$916$	36926.2	1.33196
$917$	2239.32	0.0806423
$918$	0	0
$919$	10483.3	0.376292	0.188146	−	0.982141i	$-0.439752\pi$
$919$	10483.3	0.376292	0.188146	+	0.982141i	$0.439752\pi$
$920$	7170.75	0.256970
$921$	0	0
$922$	3805.26	0.135922
$923$	46535.9	1.65953
$924$	0	0
$925$	−5733.46	−0.203800
$926$	1994.62	0.0707855
$927$	0	0
$928$	−17749.1	−0.627846
$929$	−42968.9	−1.51751	−0.758754	−	0.651377i	$-0.774191\pi$
$929$	−42968.9	−1.51751	−0.758754	+	0.651377i	$0.774191\pi$
$930$	0	0
$931$	−19990.0	−0.703702
$932$	48599.6	1.70808
$933$	0	0
$934$	−5366.27	−0.187998
$935$	−4702.70	−0.164487
$936$	0	0
$937$	−22380.4	−0.780296	−0.390148	−	0.920752i	$-0.627576\pi$
$937$	−22380.4	−0.780296	−0.390148	+	0.920752i	$0.627576\pi$
$938$	−852.972	−0.0296914
$939$	0	0
$940$	39034.0	1.35441
$941$	5654.09	0.195875	0.0979374	−	0.995193i	$-0.468776\pi$
$941$	5654.09	0.195875	0.0979374	+	0.995193i	$0.468776\pi$
$942$	0	0
$943$	33610.8	1.16068
$944$	46171.2	1.59189
$945$	0	0
$946$	−799.415	−0.0274749
$947$	−11978.1	−0.411021	−0.205510	−	0.978655i	$-0.565885\pi$
$947$	−11978.1	−0.411021	−0.205510	+	0.978655i	$0.565885\pi$
$948$	0	0
$949$	−27946.6	−0.955939
$950$	1841.70	0.0628976
$951$	0	0
$952$	2863.86	0.0974980
$953$	−550.898	−0.0187254	−0.00936271	−	0.999956i	$-0.502980\pi$
$953$	−550.898	−0.0187254	−0.00936271	+	0.999956i	$0.502980\pi$
$954$	0	0
$955$	36994.7	1.25353
$956$	1886.95	0.0638372
$957$	0	0
$958$	−114.492	−0.00386123
$959$	484.054	0.0162992
$960$	0	0
$961$	−17234.8	−0.578524
$962$	1406.89	0.0471517
$963$	0	0
$964$	−45707.7	−1.52712
$965$	−18916.4	−0.631025
$966$	0	0
$967$	34159.5	1.13598	0.567992	−	0.823034i	$-0.307721\pi$
$967$	34159.5	1.13598	0.567992	+	0.823034i	$0.307721\pi$
$968$	6628.31	0.220085
$969$	0	0
$970$	−1690.76	−0.0559659
$971$	16166.0	0.534285	0.267142	−	0.963657i	$-0.413921\pi$
$971$	16166.0	0.534285	0.267142	+	0.963657i	$0.413921\pi$
$972$	0	0
$973$	−11620.3	−0.382868
$974$	4394.67	0.144573
$975$	0	0
$976$	49097.8	1.61023
$977$	43337.2	1.41912	0.709560	−	0.704645i	$-0.248894\pi$
$977$	43337.2	1.41912	0.709560	+	0.704645i	$0.248894\pi$
$978$	0	0
$979$	3001.36	0.0979813
$980$	−24895.3	−0.811480
$981$	0	0
$982$	3134.88	0.101872
$983$	−11035.5	−0.358065	−0.179033	−	0.983843i	$-0.557297\pi$
$983$	−11035.5	−0.358065	−0.179033	+	0.983843i	$0.557297\pi$
$984$	0	0
$985$	29618.7	0.958101
$986$	4897.55	0.158184
$987$	0	0
$988$	34043.0	1.09621
$989$	38089.3	1.22464
$990$	0	0
$991$	11572.2	0.370943	0.185471	−	0.982650i	$-0.440619\pi$
$991$	11572.2	0.370943	0.185471	+	0.982650i	$0.440619\pi$
$992$	6843.86	0.219045
$993$	0	0
$994$	−3264.87	−0.104181
$995$	55129.4	1.75650
$996$	0	0
$997$	−23804.4	−0.756160	−0.378080	−	0.925773i	$-0.623416\pi$
$997$	−23804.4	−0.756160	−0.378080	+	0.925773i	$0.623416\pi$
$998$	263.143	0.00834635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.18		32
3.2	odd	2		717.4.a.d.1.15	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.15	✓	32	3.2	odd	2
2151.4.a.d.1.18		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.323742 q^{2} -7.89519 q^{4} -13.7872 q^{5} +10.6908 q^{7} -5.14594 q^{8} +O(q^{10})\)	\(q+0.323742 q^{2} -7.89519 q^{4} -13.7872 q^{5} +10.6908 q^{7} -5.14594 q^{8} -4.46348 q^{10} -6.55234 q^{11} -49.3322 q^{13} +3.46106 q^{14} +61.4956 q^{16} -52.0567 q^{17} +87.4046 q^{19} +108.852 q^{20} -2.12127 q^{22} +101.071 q^{23} +65.0858 q^{25} -15.9709 q^{26} -84.4059 q^{28} -290.605 q^{29} +112.054 q^{31} +61.0762 q^{32} -16.8529 q^{34} -147.396 q^{35} -88.0908 q^{37} +28.2965 q^{38} +70.9479 q^{40} +332.547 q^{41} +376.858 q^{43} +51.7319 q^{44} +32.7208 q^{46} +358.596 q^{47} -228.707 q^{49} +21.0710 q^{50} +389.487 q^{52} -249.863 q^{53} +90.3381 q^{55} -55.0142 q^{56} -94.0810 q^{58} +750.805 q^{59} +798.395 q^{61} +36.2767 q^{62} -472.192 q^{64} +680.151 q^{65} -246.448 q^{67} +410.998 q^{68} -47.7182 q^{70} -943.317 q^{71} +566.499 q^{73} -28.5187 q^{74} -690.076 q^{76} -70.0497 q^{77} +1184.65 q^{79} -847.849 q^{80} +107.659 q^{82} -796.105 q^{83} +717.714 q^{85} +122.005 q^{86} +33.7179 q^{88} -458.059 q^{89} -527.401 q^{91} -797.972 q^{92} +116.093 q^{94} -1205.06 q^{95} +378.798 q^{97} -74.0420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more